An extension of the internal rate of return to stochastic cash flows

The internal rate of return (IRR) is a venerable technique for evaluating deterministic cash flow streams. Part of the difficulty in extending this measure to stochastic cash flows is the lack of coherent methods for accounting for multiple or nonexistent internal rates of return in deterministic streams. Recently such a coherent theory has been developed, and we examine its implications for stochastic cash flows. We devise an extension of the deterministic IRR, which we call the stochastic rate of return on mean investment. It has significant computational and conceptual advantages over the stochastic internal rate. For instance, in the deterministic case, the standard result is that under proper conditions a cash flow stream is acceptable (in the sense of positive present value) if its internal rate exceeds the interest rate. We show that a stochastic cash flow stream is acceptable (in the sense of positive certainty equivalent expected value) if the rate of return on mean investment has a suitably defined certainty equivalent exceeding the risk-free interest rate.